Fundamental semigroups having a band of idempotents

The construction by Hall of a fundamental orthodox semigroup W B from a band B provides an important tool in the study of orthodox semigroups. We present here a semigroup S B that plays the role of W B for a class of semigroups having a band of idempotents B . Specifically, the semigroups we consider are weakly B -abundant and satisfy the congruence condition (C). Any orthodox semigroup S with E ( S )= B lies in our class. On the other hand, if a semigroup S lies in our class, then S is Ehresmann if and only if B is a semilattice. The Hall semigroup W B is a subsemigroup of S B , as are the (weakly) idempotent connected semigroups V B and U B . We show how the structure of S B can be used to extract information relating to arbitrary weakly B -abundant semigroups with (C).